# A Fast and Robust Algorithm to Count Topologically Persistent Holes in Noisy Clouds

@article{Kurlin2014AFA, title={A Fast and Robust Algorithm to Count Topologically Persistent Holes in Noisy Clouds}, author={V. Kurlin}, journal={2014 IEEE Conference on Computer Vision and Pattern Recognition}, year={2014}, pages={1458-1463} }

Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We design the algorithm to count holes that are most persistent in the filtration of offsets (neighborhoods) around given points. The input is a cloud of n points in the plane without any user-defined…

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## 18 Citations

A fast persistence-based segmentation of noisy 2D clouds with provable guarantees

- Mathematics, Computer SciencePattern Recognit. Lett.
- 2016

Skeletonisation algorithms with theoretical guarantees for unorganised point clouds with high levels of noise

- Computer SciencePattern Recognit.
- 2021

Auto-completion of Contours in Sketches, Maps, and Sparse 2D Images Based on Topological Persistence

- Computer Science2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
- 2014

This work designs a new fast algorithm to automatically complete closed contours in a finite point cloud on the plane and proves theoretical guarantees when, for a given noisy sample of a graph in the plane, the output contours geometrically approximate the original contour in the unknown graph.

Skeletonisation Algorithms for Unorganised Point Clouds with Theoretical Guarantees

- Computer ScienceArXiv
- 2019

Three algorithms that solve the data skeletonisation problem for a general cloud with topological and geometric guarantees are compared and HoPeS represents the 1-dimensional shape of a cloud at any scale by the Optimality Theorem.

A Homologically Persistent Skeleton is a fast and robust descriptor for a sparse cloud of interest points and saliency features in noisy 2 D images

- Computer Science, Environmental Science
- 2015

A classical Minimum Spanning Tree of a cloud is extended to a Homologically Persistent Skeleton, which is scale-and-rotation invariant and depends only on the cloud without extra parameters, and is geometrically stable for noisy samples around planar graphs.

A one‐dimensional homologically persistent skeleton of an unstructured point cloud in any metric space

- Environmental Science, Computer ScienceSGP '15
- 2015

A homologically persistent skeleton is defined, which depends only on a cloud of points and contains optimal subgraphs representing 1‐dimensional cycles in the cloud across all scales, and is a universal structure encoding topological persistence of cycles directly on the cloud.

Hole detection in a planar point set: An empty disk approach

- Computer ScienceComput. Graph.
- 2017

Research on a hole filling algorithm of a point cloud based on structure from motion.

- PhysicsJournal of the Optical Society of America. A, Optics, image science, and vision
- 2019

A fitting approach to fill the holes based on structure from motion (SFM) is proposed, which has been proven to be robust by experiments, and information of complex surface holes can be restored sufficiently.

Polygonal Meshes of Highly Noisy Images based on a New Symmetric Thinning Algorithm with Theoretical Guarantees

- Computer Science, PhysicsVISIGRAPP
- 2020

A new symmetric thinning algorithms to extract from such highly noisy images 1-pixel wide skeletons with theoretical guarantees that establish the state-of-the-art in extracting optimal meshes fromhighly noisy images.

A higher-dimensional homologically persistent skeleton

- Computer Science, MathematicsAdv. Appl. Math.
- 2019

## References

SHOWING 1-10 OF 10 REFERENCES

Scalar Field Analysis over Point Cloud Data

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2011

This work introduces a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated, and derives a series of algorithms for the analysis of scalar fields from point cloud data.

Image Segmentation Using Topological Persistence

- MathematicsCAIP
- 2007

The algorithm uses edge-directed topology to initially split the image into a set of regions based on the Delaunay triangulations of the points in the edge map to generate three types of regions: p-persistent regions, p-transient regions, and d-triangles.

Persistence-sensitive simplication of functions on surfaces in linear time

- Mathematics
- 2009

Persistence provides a way of grading the importance of homological features in the sublevel sets of a real-valued function. Following the definition given by Edelsbrunner, Morozov and Pascucci, an…

Data Skeletonization via Reeb Graphs

- Computer ScienceNIPS
- 2011

This paper develops a framework to extract, as well as to simplify, a one-dimensional "skeleton" from unorganized data using the Reeb graph, which is very simple, does not require complex optimizations and can be easily applied to unorganized high-dimensional data such as point clouds or proximity graphs.

The union of balls and its dual shape

- Computer ScienceSCG '93
- 1993

Efficient algorithms are described for computing topological, binatorial, and metric properties of the union of finitely many balls based on a simplicial complex dual to a certain decomposition of theunion of balls, and on short inclusion-exclusion formulas derived from this complex.

Zigzag persistent homology in matrix multiplication time

- Computer Science, MathematicsSoCG '11
- 2011

A new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions, which takes O(n3) time in the worst case.

Computational Topology - an Introduction

- Computer Science
- 2009

This book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles.

Stability of persistence diagrams

- MathematicsSCG
- 2005

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram…

Computational geometry: algorithms and applications

- Computer Science
- 1997

This introduction to computational geometry focuses on algorithms as all techniques are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems.

Data structures and network algorithms

- Computer ScienceCBMS-NSF regional conference series in applied mathematics
- 1983

This paper presents a meta-trees tree model that automates the very labor-intensive and therefore time-heavy and therefore expensive process of manually selecting trees to grow in a graph.